U.S. patent number 8,744,086 [Application Number 13/646,545] was granted by the patent office on 2014-06-03 for method and apparatus for distributing a quantum key.
This patent grant is currently assigned to The Trustees of Columbia University in the City of New York. The grantee listed for this patent is The Trustees of Columbia University in the City of New York. Invention is credited to Dirk R. Englund, Jacob Mower.
United States Patent |
8,744,086 |
Englund , et al. |
June 3, 2014 |
**Please see images for:
( Certificate of Correction ) ** |
Method and apparatus for distributing a quantum key
Abstract
A method for distributing a quantum key is provided, including
sending a first photon to a first receiver; sending a second photon
to a second receiver, the first and second photons being a pair of
time-energy entangled photons; and providing a coding scheme
comprising a plurality of time bins and a plurality of frequency
bins, wherein a combination of a time bin and a frequency bin
corresponds to a character.
Inventors: |
Englund; Dirk R. (New York,
NY), Mower; Jacob (New York, NY) |
Applicant: |
Name |
City |
State |
Country |
Type |
The Trustees of Columbia University in the City of New
York |
New York |
NY |
US |
|
|
Assignee: |
The Trustees of Columbia University
in the City of New York (New York, NY)
|
Family
ID: |
48042089 |
Appl.
No.: |
13/646,545 |
Filed: |
October 5, 2012 |
Prior Publication Data
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Document
Identifier |
Publication Date |
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US 20130089206 A1 |
Apr 11, 2013 |
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Related U.S. Patent Documents
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Application
Number |
Filing Date |
Patent Number |
Issue Date |
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61543691 |
Oct 5, 2011 |
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Current U.S.
Class: |
380/278 |
Current CPC
Class: |
H04L
9/0852 (20130101); H04L 9/0858 (20130101); H04L
2209/24 (20130101) |
Current International
Class: |
H04L
9/08 (20060101) |
Field of
Search: |
;380/278 |
References Cited
[Referenced By]
U.S. Patent Documents
Other References
"Distribuion of Time--Energy Entanglement over 100km fiber using
superconducting single--photon detectors," by Qiang, Zhang, Hiroki,
Takesue, Sae Woo Nam, Carsten Langrock, Xiuping Xie, M.M. Fejer,
Yoshihisa Yamamoto. Edward L. Ginzton Laboratory, Standford
University, Standford, California 94305. Copyright 2007 Optical
Society of America. cited by examiner .
"Frequency--bin entangled photons," by L. Olislager, J. Cussey,
A.T. Nguyen, P. Emplit, S. Massar, J.-M. Merolla, and K. Phan Huy.
Universitie' Libre de Bruxelles, Ave. F.D., Roosevelt 50, B-1050
Brussels, Belgium. (Received Jul. 8, 2009; Published Jul. 6, 2010).
cited by examiner .
Bennett, C. & Brassard, G. "Quantum cryptography: Public key
distribution and coin tossing." In Proceedings of IEEE
International Conference on Computers, Systems, and Signal
Processing, (1984) 175-179 (IEEE, New York). cited by applicant
.
Bechmann-Pasquinucci, H. & Tittel, W. Quantum cryptography
using larger alphabets. Phys. Rev. (2000) A 61, 062308. URL
http://link.aps.org/doi/10.1103/PhysRevA.61.062308. cited by
applicant .
Ali-Khan, I., Broadbent, C. J. & Howell, J. C. "Large-alphabet
quantum key distribution using energy-time entangled bipartite
states." Phys. Rev. Lett. 98, (2007) 060503. URL
http://link.aps.org/doi/10.1103/PhysRevLett.98.060503. cited by
applicant .
Barreiro, J. T., Wei, T.-C. & Kwiat, P. G. "Beating the channel
capacity limit for linear photonic superdense coding." Nature
Physics 4, (2008) 282-286. cited by applicant .
Shih, Y. H. & Alley, C. O. "New type of
Einstein-Podolsky-Rosen-Bohm Experiment Using Pairs of Light Quanta
Produced by Optical Parametric Down Conversion." Phys. Rev. Lett.
61, (1988) 2921-2924. cited by applicant .
Law, C. K. & Eberly, J. H. "Analysis and interpretation of high
transverse entanglement in optical parametric down conversion."
Phys. Rev. Lett. 92, (2004) 127903. URL
http://link.aps.org/doi/10.1103/PhysRevLett.92.127903. cited by
applicant .
Lloyd, S., Shapiro, J. H. & Wong, F. N. C. "Quantum magic
bullets by means of entanglement." J. Opt. Soc. Am. B 19, (2002)
312-318. URL http://josab.osa.org/abstract.cfm?URI=josab-19-2-312.
cited by applicant .
Hadfield, R. H. "Single-photon detectors for optical quantum
information applications." Nature Photonics 3, (2009) 696. cited by
applicant.
|
Primary Examiner: Harriman; Dant Shaifer
Attorney, Agent or Firm: Chiarini; Lisa A. Egbert, III;
Walter M. Hughes Hubbard & Reed LLP
Claims
What is claimed is:
1. A method for distributing a quantum key, the method comprising:
sending a first photon to a first receiver; sending a second photon
to a second receiver, the first and second photons being a pair of
time-energy entangled photons; and providing a coding scheme
comprising a plurality of time bins and a plurality of frequency
bins, wherein a combination of a time bin and a frequency bin
corresponds to a character.
2. The method of claim 1 wherein the first photon and the second
photon are sent via optical fiber.
3. The method of claim 1 wherein the first photon and the second
photon are sent via a photonic integrated chip.
4. The method of claim 1 wherein the first photon and the second
photon are sent through free space.
5. A method for distributing a quantum key, comprising: generating
a plurality of pairs of time-energy entangled photons; for each
pair of time-energy entangled photons, sending one photon to the
first receiver and one photon to the second receiver; and providing
a coding scheme comprising a plurality of time bins and a plurality
of frequency bins wherein each pair of one of the plurality of time
bins and one of the plurality of frequency bins corresponds to one
of a plurality of characters.
6. A method of receiving a quantum key, comprising: receiving a
first photon, the first photon being one of a pair of time-energy
entangled photons; detecting a time of arrival of the first photon;
detecting a frequency of the first photon; and determining a
character based on the detected time and frequency of the first
photon.
7. The method of claim 6 wherein the determining a character step
further comprises: assigning the detected time to a time bin;
assigning the detected frequency to a frequency bin; and
determining the character from a coding scheme based on the time
bin and the frequency bin.
8. The method of claim 6, further comprising: detecting whether an
eavesdropper has detected the first photon by determining the
frequency distribution of the first photon.
9. A method of receiving a quantum key, comprising: receiving a
plurality of photons, each of the plurality of photons being one of
a pair of time-energy entangled photons; detecting the time of
arrival of each of the plurality of photons; detecting the
frequency of each of the plurality of photons; and determining a
character based on the detected time and frequency of each of the
plurality of photons.
Description
CROSS REFERENCE TO RELATED APPLICATIONS
This application claims priority to U.S. provisional application
61/543,691 filed Oct. 5, 2011, which is incorporated by reference
in its entirety herein.
FIELD
The disclosed subject matter generally relates to integrated
optical systems. More particularly, the disclosed subject matter
relates to a new design for a photonic crystal useful for a wide
variety of applications.
BACKGROUND
Quantum key distribution (QKD) enables two parties to establish a
secure key at a distance, even in the presence of one or multiple
eavesdroppers. The key can then be used to secretly transmit
information using unconditionally secure one-time pad encryption.
Unconditional security is achieved by the laws of physics rather
than by assumptions about the computational abilities of the
eavesdropper. Various QKD protocols have been proposed and
implemented. A first protocol due employed polarization states of
photons passed between the two parties. There has been growing
interest in schemes employing photons in Hilbert spaces of high
dimension, resulting in a potentially very large alphabet size.
Different degrees of freedom have been considered, including
polarization, time, and spatial modes.
What is needed is a protocol that allows two parties to generate
their secure key at the maximum rate allowed using the time-energy
basis. Moreover, a protocol is needed that is compatible with
fiber-based dense WDM (DWDM) systems commonly used in classical
fiber communications. It is also desirable to have an extremely
compact, stable, and scalable platform for the protocol's
implementation.
SUMMARY
A method for distributing a quantum key is provided, including
sending a first photon to a first receiver; sending a second photon
to a second receiver, the first and second photons being a pair of
time-energy entangled photons; and providing a coding scheme
comprising a plurality of time bins and a plurality of frequency
bins, wherein a combination of a time bin and a frequency bin
corresponds to a character.
In some embodiments, the first photon and the second photon are
sent via optical fiber. In some embodiments, the first photon and
the second photon are sent via a photonic integrated chip. In some
embodiments, the first photon and the second photon are sent
through free space.
A method for distributing a quantum key is provided including
generating a plurality of pairs of time-energy entangled photons;
for each pair of time-energy entangled photons, sending one photon
to the first receiver and one photon to the second receiver; and
providing a coding scheme comprising a plurality of time bins and a
plurality of frequency bins wherein each pair of one of the
plurality of time bins and one of the plurality of frequency bins
corresponds to one of a plurality of characters.
A method of receiving a quantum key is provided including receiving
a first photon, the first photon being one of a pair of time-energy
entangled photons; detecting a time of arrival of the first photon;
detecting a frequency of the first photon; and determining a
character based on the detected time and frequency of the first
photon.
In some embodiments, the determining a character step further
comprises: assigning the detected time to a time bin; assigning the
detected frequency to a frequency bin; and determining the
character from a coding scheme based on the time bin and the
frequency bin.
In some embodiments, the method further includes detecting whether
an eavesdropper has detected the first photon by determining the
frequency distribution of the first photon.
A method of receiving a quantum key is provided including receiving
a plurality of photons, each of the plurality of photons being one
of a pair of time-energy entangled photons; detecting the time of
arrival of each of the plurality of photons; detecting the
frequency of each of the plurality of photons; and determining a
character based on the detected time and frequency of each of the
plurality of photons.
An apparatus for receiving a quantum key is provided including a
channel configured to receive photons; a plurality of multi-channel
filtering elements, each corresponding to a frequency and each
configured to pass photons within a range of its corresponding
frequency; and a plurality of photon detectors, each configured to
receive photons passed by one of the plurality of multi-channel
filtering elements and to send an indication of photon
detection.
BRIEF DESCRIPTION OF THE DRAWINGS
FIG. 1(a) illustrates a system using the time-coding scheme in
accordance with exemplary embodiments of the disclosed subject
matter.
FIG. 1(b) illustrates time bins in accordance with exemplary
embodiments of the disclosed subject matter.
FIG. 1(c) illustrates a further system using the time-coding scheme
in accordance with exemplary embodiments of the disclosed subject
matter.
FIG. 1(d) illustrates time bins in accordance with exemplary
embodiments of the disclosed subject matter.
FIG. 2(a) illustrates Franson interferometers in accordance with
exemplary embodiments of the disclosed subject matter.
FIG. 2(b) illustrates the coincidence counting probability P.sub.C
displayed versus .delta.t in accordance with exemplary embodiments
of the disclosed subject matter.
FIG. 3(a) illustrates mutual information as a function of detector
jitter, .sigma..sub.det, normalized to .sigma..sub.bin in
accordance with exemplary embodiments of the disclosed subject
matter.
FIG. 3(b) illustrates mutual information as a function of frequency
channels, n.sub.f in accordance with exemplary embodiments of the
disclosed subject matter.
FIG. 4 illustrates a further Franson interferometer in accordance
with exemplary embodiments of the disclosed subject matter.
DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS
It is understood that the subject matter described herein is not
limited to particular embodiments described, as such may, of
course, vary. It is also understood that the terminology used
herein is for the purpose of describing particular embodiments
only, and is not intended to be limiting, since the scope of the
present subject matter is limited only by the appended claims.
Where a range of values is provided, it is understood that each
intervening value between the upper and lower limit of that range,
and any other stated or intervening value in that stated range, is
encompassed within the disclosed subject matter.
Unless defined otherwise, all technical and scientific terms used
herein have the same meaning as commonly understood by one of
ordinary skill in the art to which this disclosed subject matter
belongs. Although any methods and materials similar or equivalent
to those described herein can also be used in the practice or
testing of the present disclosed subject matter, this disclosure
may specifically mention certain exemplary methods and
materials.
All publications mentioned in this disclosure are, unless otherwise
specified, incorporated by reference herein for all purposes,
including, without limitation, to disclose and describe the methods
and/or materials in connection with which the publications are
cited.
The publications discussed herein are provided solely for their
disclosure prior to the filing date of the present application.
Nothing herein is to be construed as an admission that the present
disclosed subject matter is not entitled to antedate such
publication by virtue of prior invention. Further, the dates of
publication provided may be different from the actual publication
dates, which may need to be independently confirmed.
As used herein and in the appended claims, the singular forms "a,"
"an" and "the" include plural referents unless the context clearly
dictates otherwise.
Nothing contained in the Abstract or the Summary should be
understood as limiting the scope of the disclosure. The Abstract
and the Summary are provided for bibliographic and convenience
purposes and due to their formats and purposes should not be
considered comprehensive.
As will be apparent to those of skill in the art upon reading this
disclosure, each of the individual embodiments described and
illustrated herein has discrete components and features which may
be readily separated from or combined with the features of any of
the other several embodiments without departing from the scope or
spirit of the present disclosed subject matter. Any recited method
can be carried out in the order of events recited, or in any other
order that is logically possible.
Reference to a singular item includes the possibility that there
are plural of the same item present. When two or more items (for
example, elements or processes) are referenced by an alternative
"or," this indicates that either could be present separately or any
combination of them could be present together, except where the
presence of one necessarily excludes the other or others.
As summarized above and as described in further detail below, in
accordance with the various embodiments of the present invention, a
large-alphabet protocol is described herein referred to as a
`Wavelength Division Multiplexed Quantum Key Distribution`
(WDM-QKD) that employs time-energy entangled photon pairs. Given a
certain photon flux n and channel bandwidth .DELTA..OMEGA., WDM-QKD
allows two parties, e.g., "Alice and Bob," to generate their secure
key at the maximum rate allowed using the time-energy basis.
WDM-QKD forms the quantum-cryptography analog of present-day WDM
systems and is compatible with fiber-based dense WDM (DWDM) systems
commonly used in classical fiber communications. An implementation
of WDM-QKD is described herein as both a modern fiber optic network
and in a photonic integrated chip (PIC). The latter provides an
extremely compact, stable, and scalable platform for the protocol's
implementation.
WDM-QKD enables Alice and Bob to generate their shared key at a
maximum rate of n log.sub.2(.DELTA..OMEGA./n) at
log.sub.2(.DELTA..OMEGA./n) bits per photon (bpp), using
present-day single photon counters, WDM equipment, and simple
components such as beam splitters or directional couplers. The
information per photon per bandwidth is log.sub.2
(.DELTA..OMEGA./n)/.DELTA..OMEGA..
QKD protocols derive security from the fact that a measurement
changes an unknown state if measured in a conjugate basis. If Alice
and Bob employ two conjugate bases, they can detect a measurement
by a third party eavesdropper, "Eve," including quantum
nondemolition measurements. In the protocol described herein, the
conjugate bases are time and energy. Alice and Bob make time and
energy measurements using an extended Franson interferometer.
Assuming 400 wavelength channels with detectors having 40 ps
jitter, it is expected that for a bandwidth .DELTA..OMEGA.=10 nm
around 1550 nm, Alice and Bob can generate a key at a maximum of 10
Tera bits per second (Tbps) at 1 bit per photon (bpp); or at 10 bpp
and 100 Gbps; or at 200 Mbps at 20 bpp.
Time-energy entangled photon pairs with correlation time
.sigma..sub.cor generated by spontaneous parametric down conversion
(SPDC) were considered herein, assuming a pump field at frequency
.omega..sub.p with coherence time .sigma..sub.coh. The photon pair
wave function can be written as
|.PSI.=.intg..sub.-.infin..sup..infin..intg..sub.-.infin..sup..infin..psi-
.(t.sub.A,t.sub.B)|t.sub.A,t.sub.B,o.sub.A,o.sub.Bdt.sub.Adt.sub.B
(1) |t.sub.A,t.sub.B,o.sub.A,o.sub.B={circumflex over
(.alpha.)}.sub.o.sub.A.sup..dagger.(t.sub.A(t.sub.A){circumflex
over (.alpha.)}.sub.o.sub.B.sup..dagger.(t.sub.B)|0
.psi.(t.sub.A,t.sub.B).varies.e.sup.-(t.sup.A.sup.-t.sup.B.sup.).sup.2.su-
p./4.sigma..sup.car.sup.2e.sup.-(t.sup.A.sup.+t.sup.B.sup.).sup.2.sup./16.-
sigma..sup.car.sup.2e.sup.iw.sup.p.sup./2(t.sup.A.sup.+t.sup.B.sup.)
The creation operator {circumflex over
(.alpha.)}.sub.o.sub.i.sup..dagger.(t.sub.j) denotes creation in
spatial mode o.sub.i at time t.sub.j. If the correlation time of
the photons is sufficiently smaller than the time bin duration,
Alice and Bob can establish a secret key using the correlated
photon arrival time. A schematic of this protocol is shown in FIG.
1. The continuous biphoton wave function can be discretized as a
sum over time bins |.sigma..sub.bin.sup.i of duration
.sigma..sub.bin as
.PSI..infin..infin..times..infin..infin..times..times..sigma..sigma..time-
s..times..intg..times..times..sigma..times..sigma..times..intg..times..tim-
es..sigma..times..times..times..sigma..times..times..times..psi..function.-
.times.d.times.d ##EQU00001##
The probability that Alice and Bob project into time bins
.sigma..sub.bin.sup.i and .sigma..sub.bin.sup.j is therefore
p.sup.i,j=.sigma..sub.bin.sup.i, .sigma..sub.bin.sup.j|
.psi.|.sup.2=|G.sup.i,j|.sup.2. Detector jitter of magnitude
.sigma..sub.det influences the fidelity of this projection, so that
roughly .sigma..sub.bin>.sigma..sub.det for reliable
communication. Jitter therefore reduces the maximum number of
characters to .sigma..sub.coh/.sigma..sub.det from the maximum
allowed, given by the Schmidt number,
K.apprxeq..sigma..sub.coh/.sigma..sub.cor. The fastest single
photon detectors provide .sigma..sub.det.apprxeq.40 ps, whereas
.sigma..sub.cor can be on the order of 10 fs-1 ps.
FIG. 1(a) illustrates a system 100 using the time-coding scheme. A
strong laser (not shown) pumps a nonlinear crystal. Photons pairs
are generated by SPDC 102 and sent across channels of equal length
to detectors 106 of Alice's computer 108 and Bob's computer 110 who
measure their arrival times. FIG. 1(b) illustrates time bins agreed
upon by Alice and Bob over a public channel. If a photon pair is
detected in a given time bin, then that character is shared between
Alice and Bob. FIG. 1(c) illustrates a system 200 in which photons
pairs are also generated by SPDC 202. Alice's computer 208 and
Bob's computer 210 include a grating, dispersive element or
multi-channel filter 212 before their detectors 206 to obtain
frequency information. FIG. 1(d) illustrates time bins.
For technological reasons, detector jitter is unlikely to approach
the sub-ps regime in the near future. To overcome the technological
mismatch between photon correlation time and detector jitter, the
protocol described herein utilizes the circumstance that photons
are not only entangled in time, but also in frequency. The biphoton
wave function in the frequency domain, |.PSI..sub.F=FT.sub.2|.PSI.,
where FT.sub.2 denotes the two-dimensional Fourier transform.
Therefore,
|.PSI..sub.F=.intg..intg.(.omega..sub.A,.omega..sub.B)|.omega..sub.A,.ome-
ga..sub.B,o.sub.A,o.sub.Bd.omega..sub.Ad.omega..sub.B, where
.psi.(.omega..sub.A,.omega..sub.B).alpha.exp[-.sigma..sub.cor.sup.2/4(.om-
ega..sub.A-.omega..sub.B).sup.2]exp[-.sigma..sub.coh.sup.2/4(.omega..sub.A-
+.omega..sub.B-.omega..sub.p).sup.2].
States of this form show non-local frequency correlations. Thus, if
Alice measures a frequency .omega..sub.A on one photon, then Bob
must measure a frequency
.omega..sub.B.apprxeq..omega..sub.p-.omega..sub.A on the other,
where .omega..sub.B=.omega..sub.p-.omega..sub.A for
.sigma..sub.coh.fwdarw..infin.. If both Alice and Bob place
multi-channel filtering elements before their detectors (as shown
in FIG. 1(c)), then they can collect frequency information in
addition to timing information. This filtering can be modeled as
Gaussian projections onto discrete output spatial modes
|.zeta..sup.i with frequency bandwidth .delta.v and center
frequency v.sub.i. Such filtering increases the correlation time of
the photon pair, but this can be kept lower than the detector
jitter to minimize the loss of timing information. Alternatively,
one can increase the time bin duration .sigma..sub.bin to allow for
smaller .delta.v. Optimization of these parameters given
limitations of the source and transmission channel are described
below.
The QKD protocol is secure as discussed herein. Measurements of the
frequency and creation time of the biphoton packet disturb its wave
function in these bases; temporal positive operator valued
measurements (POVM) reduce the coherence time while frequency POVMs
increase the correlation time as discussed in greater detail
below.
Alice and Bob can check the security of their channel by testing
for changes in the correlation and coherence times. They use an
extended Franson interferometer (eFI) 300/301, depicted in FIG.
2(a), in which photons pairs are also generated by SPDC 302. eFI
300/301 include, e.g., beam splitters BS 326, partial beam
splitters PBS 328, and detectors 306. Measurement in Alice's and
Bob's arm of the eFI are described by the annihilation operators
for the long and short paths, with respective times t.sub.L.sup.i
and t.sub.S.sup.i, and i.epsilon.A, B denoting Alice and Bob:
aA(tA)=1/ {square root over
(2)}[aA(t.sub.L.sup.A)+aA(t.sub.S.sup.A)] and aB(tB)=1/ {square
root over (2)}[aB(t.sub.L.sup.B)+aA(t.sub.S.sup.B)].
FIG. 2(a) illustrates for Bob's eFI 301, the short arm is switched
between position A and B to achieve lengths .delta.t.sub.1 and
.delta.t.sub.2. This allows determination of both and
.sigma..sub.cor so that weak frequency and time measurements on the
photon pair can be detected. FIG. 2(b) illustrates the coincidence
counting probability P.sub.C displayed versus .delta.t. For
example, at position A, .delta.t=0 and P.sub.c=1.0.
These times are redefined using standard notation, where
.DELTA.t=(t.sub.L.sup.A-t.sub.S.sup.A) and
.delta.t=(t.sub.L.sup.A-t.sub.S.sup.A)-(t.sub.L.sup.B-t.sub.S.sup.B).
A selection is made to use .DELTA.t,
.DELTA.t-.delta.t>>.sigma..sub.cor in order to avoid single
photon interference between long and short paths of a single arm of
the eFI. Selection of two different values, .delta.t.sub.1,2, is
allowed by placing a switch in the long path of one arm of the eFI.
The parameter .delta.t.sub.1,2 is then scanned to provide
phase-dependent coincidence counting measurements. Alice and Bob
measure the coincidence counting probability P.sub.C, which
evaluates to P.sub.c.varies.1/2+1/2 cos
[.omega.(2.DELTA.t-.delta.t)]e.sup.-.delta.t.sup.2.sup./8.sigma..sup.coh.-
sup.2e.sup.-.DELTA.t.sup.2.sup./8.sigma..sup.coh.sup.2 with
visibility
V=e.sup..delta.t.sup.2.sup./8.sigma..sup.cor.sup.2e.sup.-.DELTA.t.sup.2.s-
up./8.sigma..sup.coh.sup.2. The correlation time and coherence time
can be deduced from two visibility measurements V.sub.1 and V.sub.2
using .delta.t.sub.1 and .delta.t.sub.2, respectively. These
extrapolated values .sigma..sub.coh.sup.E' and
.sigma..sub.cor.sup.E', are given by
.sigma.'.times..delta..times..times..delta..times..times..times..times..t-
imes..times..sigma.'.times..DELTA..times..times..function..delta..times..t-
imes..delta..times..times..delta..times..times..times..times..times..delta-
..times..times..times..times..times. ##EQU00002## The parameters
.delta.t and .delta.t.sub.2 are set so that
.delta.t-.delta.t.sub.2.apprxeq..sigma..sub.cor and
|.delta.t.noteq.|.delta.t.sub.2| to sample the Franson curve at
different points.
FIGS. 2(a) and 2(b) illustrate the choice of .delta.t.sub.1=0 and
.delta.t.sub.2.apprxeq..sigma..sub.cor. Using
(.sigma..sub.coh.sup.E).sup.2=1/[(.sigma..sub.coh.sup.E').sup.-2-.sigma..-
sub.coh.sup.-2] and
(.sigma..sub.cor.sup.E).sup.2=1/[(.sigma..sub.cor.sup.E').sup.-2-.sigma..-
sub.cor.sup.-2] derived from this measurement, the bound on Eve's
information per photon is
I.sub.E.ltoreq.log.sub.2(.sigma..sub.coh/.sigma..sub.coh.sup.E)+log.sub.2-
(.sigma..sub.cor.sup.E/.sigma..sub.cor), which is the sum of her
information obtained from temporal and frequency measurements.
While all change in the wave function can be due to Eve's action,
the analysis does not assume that Alice and Bob have perfect
detectors or receive unaltered photon pairs. A more pessimistic
calculation is required for the mutual information between Alice
and Bob.
The probability of Alice and Bob measuring in frequency channels i
and j and time bins k and l, is
p.sup.ijkl=|.zeta..sub.A.sup.i,.zeta..sub.B.sup.j,.sigma..sub.bin,A.sup.k-
,.sigma..sub.bin,B.sup.l|.PSI.|.sup.2=|G.sup.ijkl|.sup.2. These
coefficients form a joint probability density function which can be
used to compute the Shannon entropy S=.SIGMA..sub.ijklp.sup.ijkl
log p.sup.ijkl and mutual information I (A, B)=S(A)+S(B)-S(A,
B).
.times..function..times..times..GAMMA..times..times..GAMMA..times..times.-
.times..times..times..GAMMA..intg..times..times..sigma..times..times..time-
s..sigma..times..times..times..intg..infin..infin..times..function..intg..-
times..times..delta..times..times..times..delta..times..times..times..intg-
..infin..infin..times..psi..function..omega..omega..times.d.omega..times.d-
.omega..times.d.times.d ##EQU00003##
Detector jitter is included by operating a Gaussian spreading
function on the biphoton state as {circumflex over
(.sigma.)}.sub.det=.intg.e.sup.-t.sup.x.sup.2.sup./2.sigma..sup.det.sup.2-
|tt+t.sub.x|dt.sub.x. FIG. 3 shows the mutual information as a
function of the detector jitter normalized to the time bin
duration, as discussed hereinbelow. The joint probability density
function P of the new state, |.PSI.'={circumflex over
(.sigma.)}.sub.det,A{circumflex over
(.sigma.)}.sub.det,B{E}S.sub.BS.sub.A|.PSI..sub.F, incorporating
the set of Eve's POVMs, {E}, has elements
p.sup.ijkl=|.zeta..sub.A.sup.k.zeta..sub.B.sup.l,.sigma..sub.bin,A.sup.i,-
.sigma..sub.bin,B.sup.j|.PSI.'|.sup.2.
FIG. 3(a) illustrates mutual information as a function of detector
jitter, .sigma..sub.det, normalized to .sigma..sub.bin. FIG. 3(b)
illustrates mutual information as a function of frequency channels,
n.sub.f. The individual frequency channels have bandwidth
.delta.v=.DELTA..omega./n.sub.f and spacing 4.delta.v. Increasing
the number of frequency channels increases the bits per photon,
until the filter photon temporal bandwidth approaches
.sigma..sub.bin.
The number of time- and frequency-encoded bits given a photon
budget and channel bandwidth can be optimized. For example, the
photon budget is specific to the photon pair source and is given by
the maximum emission rate, R.sub.v. The allotted channel frequency
bandwidth is .DELTA..OMEGA.. The frequency bandwidth of an
individual photon pair source is
.DELTA..omega..varies.1/.sigma..sub.cor, so the number of sources
to fill the channel bandwidth is N=.DELTA..OMEGA./.DELTA..omega..
The communication rate, R.sub.C, in bits per second is
therefore
.times..function..DELTA..times..times..omega..delta..times..times..times.-
.sigma. ##EQU00004## using .sigma..sub.bin and .delta.v defined
above.
The often limited photon budget for quantum communication makes
high-dimensional encoding desirable. However, achieving the limit
on this dimensionality in the time domain using energy-time
entangled photon pairs requires detectors with sub-ps timing jitter
and resolution. By invoking conjugate frequency correlations, a
protocol to approach this fundamental limit has been developed
using current detectors and existing telecom networks. The
conjugate nature of time and energy encoding means that one can
trade frequency for temporal bits (and vice versa) to minimize the
effect of channel distortion such as nonlinear frequency conversion
and dispersion, in addition to optimizing over transmission rate
and channel bandwidth.
Gaussian filtering functions are assumed with center frequencies
v.sub.i and bandwidth .delta.v, which project onto spatial modes
|.zeta..sub.i. This is summarized in the following relation, where
the total filtering operator S is taken as a sum over the
individual channel filters S.sup.i.
.times..times..intg..infin..infin..times..function..omega..times..delta..-
times..times..times..omega..zeta..times..omega..times.d.omega.
##EQU00005##
The Franson interference derived hereinabove assumes lossless
propagation through the interferometer. However, this assumption is
not valid in photonic integrated chips or fiber networks. Loss can
be taken into account by placing a beam splitter in the long path
of the Franson, which couples the waveguide mode with a vacuum mode
(see FIG. 4). Working in the Heisenberg construction, evolving the
annihilation operator through the virtual loss beam splitter and
the two Franson beam splitters. The matrix for beam splitters 1 and
2, which leave the third mode undisturbed is given by
##EQU00006## where i .epsilon. 1,2. The virtual loss beam splitter
is given by
##EQU00007##
The resulting annihilation operators are then
a.sub.A(t.sub.A)=C.sub.1a(t)+C.sub.2a(t-.DELTA.t) and
a.sub.B(t.sub.B)=C.sub.1a(t)+C.sub.2a(t-.DELTA.t-.delta.t),
disregarding the vacuum term, which will not affect coincidence
counting. C.sub.1= {square root over (r.sub.1)} {square root over
(r.sub.2)} and C.sub.2= {square root over (1-r.sub.1)} {square root
over (1-r.sub.2)} {square root over (r.sub.L)}. For maximum
visibility, C.sub.1=C.sub.2, so
.times..times..times. ##EQU00008##
This is plotted. For, r.sub.1=r.sub.2=1/2, the visibility
simplifies to
.times.e.times..DELTA..times..times..tau..alpha.e.times..DELTA..times..ti-
mes..tau..alpha..times.e.delta..times..times..times..sigma..times.e.DELTA.-
.times..times..times..sigma. ##EQU00009## where .tau..sub..alpha.
is the photon lifetime in the waveguide due to loss.
Detector jitter refers to the added uncertainty in the photon
detection time of some stimulus, purely a result of detector
electronics. Superconducting nanowire single photon detectors and
InGaAs APDs both exhibit jitter of roughly 30 to 40 ps. Detector
jitter is modeled as a Gaussian projection, {circumflex over
(.sigma.)}.sub.det=.intg.e.sup.-t.sup.x.sup.2.sup./2.sigma..sup.det.sup.2-
|tt+t.sub.x|dt.sub.z. The jitter profile is not truly Gaussian and
can be quite asymmetric, however this model allows for first-order
analysis. If this is operated on both Alice and Bob's photons,
assuming Eq. (1),
.sigma..times..sigma..times..PSI..varies..intg..infin..infin..times..intg-
..infin..infin..times..function..times..sigma..times..sigma..times..functi-
on..times..sigma..times..sigma..times..times.d.times.d
##EQU00010##
Since .sigma..sub.coh.sigma..sub.det, the most important effect of
jitter is to increase the observed correlation time roughly from
.sigma..sub.cor to .sigma..sub.det. This can have a significant
effect on the mutual information between Alice and Bob if
.sigma..sub.det is on the order of .sigma..sub.bin, as shown in
FIG. 3.
The case of a single eavesdropper measuring a single photon of the
photon pair is considered herein. Eve's temporal measurement is a
POVM that can be written as a Gaussian filtering function:
{circumflex over
(E)}.sub.t=.intg..sub.-.infin..sup..infin.e.sup.-t.sup.2.sup./2(.sigma..s-
up.coh.sup.E'.sup.).sup.2|tt|dt (13)
Following, the amplitude function
.psi.(t.sub.A,t.sub.B).alpha.exp[-(t.sub.A-t.sub.B).sup.2/4.sigma..sub.co-
r.sup.2]exp[-t.sub.A.sup.2/4.sigma..sub.coh.sup.2] for
.sigma..sub.coh>>.sigma..sub.cor. Therefore
.PSI..times..PSI..varies..intg..infin..infin..times..intg..infin..infin..-
times..function..function..times..sigma..times..sigma.'.times..times..func-
tion..times..sigma..times..times.d.times.d ##EQU00011## so the
coherence time of the biphoton packet is strongly influenced by
Eve's filtering bandwidth for
.sigma..sub.coh.sup.E>>.sigma..sub.coh, which gives the bound
on her timing information.
Similarly, a weak frequency POVM is defined,
E.sub..omega.=.intg..sub.-.infin..sup..infin.e.sup.-(.sigma..sup.cor.sup.-
E'.sup.).sup.2.sup.(.omega.-.omega..sup.p.sup./2).sup.2|.omega..omega.|d.o-
mega. (15)
For 1/.sigma..sub.cor>>1/.sigma..sub.coh, and ignoring the
spatial modes, |.PSI..sub.F can be written as
|.PSI..sub.F.apprxeq..intg..intg.exp[-.sigma..sub.cor.sup.2/4(2.omega..su-
b.A-.omega..sub.p).sup.2]exp[-.sigma..sub.coh.sup.2(.omega..sub.A+.omega..-
sub.B-.omega..sub.p).sup.2]|.omega..sub.A,.omega..sub.Bd.omega..sub.Ad.ome-
ga..sub.B. (16) {circumflex over
(E)}.sub..omega.|.PSI..sub.P.apprxeq..intg..intg.exp[-(.sigma..sub.cor.su-
p.2/4+(.sigma..sub.cor.sup.E').sup.2/4)(2.omega..sub.A-.omega..sub.p).sup.-
2].times.exp[-.sigma..sub.coh.sup.2(.omega..sub.A+.omega..sub.B-.omega..su-
b.p).sup.2]|.omega..sub.A,
.omega..sub.Bd.omega..sub.Ad.omega..sub.B. (17) Thus, Eve projects
the biphoton pair onto a narrower frequency distribution.
As will be apparent to those of skill in the art upon reading this
disclosure, each of the individual embodiments described and
illustrated herein has discrete components and features which may
be readily separated from or combined with the features of any of
the other several embodiments without departing from the scope or
spirit of the present disclosed subject matter.
* * * * *
References